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ARTICLE IN PRESSG ModelRE-6260; No. of Pages 14

Precision Engineering xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Precision Engineering

jo ur nal ho me p age: www.elsev ier .com/ locate /prec is ion

new methodology to design multi-sensor networks for distributedarge-volume metrology systems based on triangulation

. Maisano ∗, L. Mastrogiacomoolitecnico di Torino (DIGEP), Torino 10129, Italy

r t i c l e i n f o

rticle history:eceived 23 March 2015eceived in revised form 29 May 2015ccepted 1 July 2015vailable online xxx

eywords:arge-volume metrologyistributed measuring systemulti-sensor network

riangulationetwork designetwork density

a b s t r a c t

Distributed Large-Volume Metrology (LVM) systems are mainly used for industrial applications concern-ing assembly and dimensional verification of large-sized objects. These systems generally consist of a setof network devices, distributed around the measurement volume, and some targets to be localized, incontact with the measured object’s surface or mounted on a hand-held probe for measuring the points ofinterest. Target localization is carried out through several approaches, which use angular and/or distancemeasurements by network devices.

This paper presents a new methodology to support the design of networks of devices, for distributedLVM systems based on triangulation (i.e., systems in which network devices perform angular measure-ments only). It is assumed that these systems use multi-sensor networks including two typologies ofdevices: some are accurate but expensive and other ones are less accurate but cheaper. The goal of themethodology is establishing a link between the following parameters: (i) density of network devices,(ii) mix between the two typologies of network devices, (iii) measurement uncertainty, and (iv) cost.The methodology allows to estimate the most appropriate density and mix between the two typologies
of network devices, so that the distributed LVM system is conforming with the required measurementuncertainty and cost.
The methodology relies on a large number of simulated experiments, defined and implemented usinga dedicated routine; feasibility and practicality is tested by preliminary experiments on a multi-sensorphotogrammetric system, developed at Politecnico di Torino—DIGEP.

© 2015 Elsevier Inc. All rights reserved.

. Introduction

The use of distributed systems for applications in the field ofarge-Volume Metrology (LVM) is more and more diffused and con-olidated [1]. Typical industrial applications concern assembly andimensional verification of large-sized mechanical components, inhich levels of accuracy of several tenths of millimetre are tol-

rated [2,3]. The reason behind the diffusion of distributed LVMystems is simple: arranging a portable measuring system aroundhe object to be measured is often more practical than the vice versa4,5].

In general, distributed LVM systems consist of: (i) a set of net-ork devices, distributed around the object to be measured, (ii)

Please cite this article in press as: Maisano D, Mastrogiacomo L. A nelarge-volume metrology systems based on triangulation. Precis Eng (2

ome targets to be localized, generally in contact with the measuredbject’s surface, or mounted on a hand-held probe for measur-ng the points of interest, and (iii) a centralized data processing

∗ Corresponding author. Tel.: +39 0110907281.E-mail address: [email protected] (D. Maisano).

ttp://dx.doi.org/10.1016/j.precisioneng.2015.07.001141-6359/© 2015 Elsevier Inc. All rights reserved.

unit (DPU), which receives local measurement data from networkdevices.

The localization of targets is carried out through three possibleapproaches:

• Triangulation, using the angles subtended by targets, with respectto network devices;

• Multilateration, using the distances between targets and networkdevices;

• Hybrid techniques, based on the combined use of angles and dis-tances between targets and network devices.

In this paper we deal exclusively with distributed LVM systemsbased on triangulation; Fig. 1 provides a schematic representa-tion of these systems: each ith network device (Di) is associatedwith a local coordinate system oixiyizi and is able to perform local

w methodology to design multi-sensor networks for distributed015), http://dx.doi.org/10.1016/j.precisioneng.2015.07.001

angular measurements with respect to a target P; the aim of themeasurement is localizing P, determining its spatial coordinates inthe global Cartesian coordinate system OXYZ. In Fig. 1, four net-work devices (i.e., D1 to D4) are involved in the measurement, as it

dx.doi.org/10.1016/j.precisioneng.2015.07.001


http://www.sciencedirect.com/science/journal/01416359

http://www.elsevier.com/locate/precision

mailto:[email protected]


ARTICLE IN PRESSG ModelPRE-6260; No. of Pages 14

2 D. Maisano, L. Mastrogiacomo / Precision Engineering xxx (2015) xxx–xxx

c distr

iIto�

tthmtlt(dcp

t

•

•

octnvo

wno

da

Fig. 1. Schematic representation of a generi

s assumed that they all include P in their range of measurement.n general, the number of devices involved in the localization of aarget depends on their mutual positioning/orientation and rangef measurement. Each ith network device measures two angles, i.e.,i and ϕi, described in Section 2.

Network devices may differ in technology, measurement uncer-ainty, range of measurement and cost. For example, opticalheodolites have uncertainty on angular measurements of a fewundredths of a degree, range of measurement of several tens ofeters (with some constraints regarding angles) and cost of a few

housand D ; the relatively recent rotary-automatic laser theodo-ites (R-LATs) have uncertainty on angular measurements of a fewhousandths of a degree [6], range of measurement of about 20 mwith some constraints regarding angles) and cost around 50,000D ;ata related to photogrammetric cameras may fluctuate signifi-antly, depending on their technical/metrological features, such asixel resolution and frame rate.

When designing a network of devices for a distributed LVM sys-em, two of the most important factors to consider are:

network density (i.e., number of network devices per surface unit),since the uncertainty in target localization tends to decrease withthe number of devices, which can “see” the target [7];uncertainty of network devices in measuring the angles subtendedby targets. In general, the more technologically advanced andexpensive the network devices, the lower their uncertainty inlocal angular measurements and, hence, that in target localiza-tion.

These factors could be both optimized by using dense networksf very accurate1 devices, although this solution may result in highost, in contrast to budget sustainability. A reasonable compromiseo achieve good results, while limiting cost, is using multi-sensor2


etworks, which combine: (i) a relatively large number of notery accurate but cheap devices, for obtaining a good coveragef the measurement volume, and (ii) a relatively low number of

1 The adjective “accurate” is used in a broad sense, denoting the ability of a net-ork device to perform angular measurements with relatively low uncertainty; it isot necessarily related to the specific definition from the International Vocabularyf Metrology (VIM) [18].2 The adjective “multi-sensor” indicates that networks include devices, which –

espite they all perform angular measurements – may differ in terms of technology,ccuracy, range of measurement and cost.

ibuted LVM system, based on triangulation.

accurate but expensive devices, for reducing the uncertainty intarget localization.

Unfortunately, defining the optimal network density and mixbetween accurate and less-accurate devices is a difficult task, dueto the complexity of the target localization problem and the factthat it can be influenced by several parameters related to net-work devices (e.g., range of measurement; uncertainty in angularmeasurement; uncertainty in their location/orientation, due to thecalibration process, etc.) [5].

The aim of this paper is introducing a new supporting method-ology to design multi-sensor networks for distributed LVM systemsbased on triangulation, assuming that these networks include twotypologies of devices: some are accurate but expensive and otherones are less accurate but cheaper. The goal of the proposedmethodology is establishing a link between four parameters: den-sity of network devices (ı), percentage of accurate devices (pA),measurement uncertainty, and cost. This will be done through alarge number of simulated experiments, in which the former twoparameters (ı and pA) are varied and their influence on the lattertwo parameters is analyzed. In these simulated experiments, tar-get localization is modelled through a consolidated mathematicalmodel, which can be adapted to multi-sensor networks. In practicalterms, the proposed methodology allows to estimate the optimal ıand pA, so that the whole system is conforming with the requiredmeasurement uncertainty and cost.

The remainder of this paper is organized into three sections. Sec-tion 2 provides some basic concepts concerning the problem of thelocalization by triangulation and the mathematical model, which isused in the proposed methodology. Section 3 is divided into threeparts: the first one describes the methodology, focusing on themulti-sensor network modelling; the second one provides a practi-cal application to a distributed LVM system, which adopts two typesof photogrammetric cameras; the third part checks the plausibilityof the results of the previous application, on the basis of preliminaryexperiments, carried out at Politecnico di Torino—DIGEP. Section 4summarizes the original contributions of this research, focussingon its implications, limitations and possible future developments.

2. The triangulation problem


Fig. 1 depicts a distributed LVM system based on triangula-tion, consisting of a number of network devices positioned aroundthe measurement volume. OXYZ is a global Cartesian coordinatesystem. Each ith device (Di) has its own spatial position and


ARTICLE ING ModelPRE-6260; No. of Pages 14

D. Maisano, L. Mastrogiacomo / Precision

Fig. 2. For a generic network device (D ), two angles – i.e., � (azimuth) and ϕ (ele-vo

ow

d

X

Rt

R in �i

in �i

w�wad

i

npcsms

tatadszje

they can be replaced with appropriate estimates: � and ϕ , resulting

i i i

ation) – are subtended by a line joining the point P (to be localized) and the origini of the local coordinate system oixiyizi .

rientation, and a local coordinate system, oixiyizi, roto-translatedith respect to OXYZ.

A general transformation between a local and the global coor-inate system is:

= Rixi + X0i⇒

⎡⎣X

Y

Z

⎤⎦ =

⎡⎢⎣

r11ir12i

r13i

r21ir22i

r23i

r31ir32i

r33i

⎤⎥⎦⎡⎢⎣

xi

yi

zi

⎤⎥⎦+

⎡⎢⎣

X0i

Y0i

Z0i

⎤⎥⎦ . (1)

i is a rotation matrix, which elements are functions of three rota-ion parameters:

i =

⎡⎢⎣

cos �i cos �i − cos �i sin �i

cos ωi sin �i + sin ωi sin �i cos �i cos ωi cos �i − sin ωi sin �i s

sin ωi sin �i − cos ωi sin �i cos �i sin ωi cos �i + cos ωi sin �i s

here ωi represents a counterclockwise rotation around the xi axis;i represents a counterclockwise rotation around the new yi axis,hich was rotated by ωi; �i represents a counterclockwise rotation

round the new zi axis, which was rotated by ωi and then �i; foretails, see [8].

X0i=[X0i

, Y0i, Z0i

]Tare the coordinates of the origin of oixiyizi,

n the global coordinate system OXYZ.The (six) location/orientation parameters related to each ith

etwork device (i.e., X0i, Y0i

, Z0i, ωi, �i,�i) are treated as known

arameters, since they are measured in an initial calibration pro-ess. The calibration process, which may vary depending on thepecific technology of the measuring system, generally includesultiple measurements of calibrated artefacts, within the mea-

urement volume [9].The point to be localized is P = [X, Y, Z]T, e.g., the position of the

arget in Fig. 1. From the local perspective of each ith device, twongles – i.e., �i (azimuth) and ϕi (elevation) – are subtended byhe line passing through P and a local observation point, which wessume as coincident with the origin oi = [0, 0, 0]T of the local coor-inate system (see Fig. 2). Precisely, ϕi describes the inclination ofegment oiP with respect to the plane xiyi (with a positive sign wheni > 0), while �i describes the counterclockwise rotation of the pro-ection (oiP′) of oiP on the xiyi plane, with respect to the xi axis. Forach ith local coordinate system, the following relationships hold:

�i = tan−1 yi

x

⎧⎪⎨ if xi ≥ 0 then − �

2≤ �i ≤ �

2


i ⎪⎩ if xi < 0 then�

2< �i <

3�

2ϕi = sin−1 zi

oiP

{−�

2≤ ϕi ≤ �

2

. (3)

PRESS Engineering xxx (2015) xxx–xxx 3

sin �i

− sin ωi cos �i

cos ωi cos �i

⎤⎥⎦ , (2)

Given that:

tan �i = sin �i

cos �i(4)

and

oiP = oiP′

cos ϕi= xi/ cos �i

cos ϕi= xi

cos �i · cos ϕi, (5)

Eq. (3) can be reformulated as:

xi · sin �i − yi · cos �i = 0

xi · sin ϕi − zi · cos �i · cos ϕi = 0.(6)

In matrix form, Eq. (6) becomes:

M ixi =[

sin �i − cos �i 0

sin ϕi 0 − cos �i · cos ϕi

]·

⎡⎢⎣

xi

yi

zi

⎤⎥⎦ = 0 (7)

The system of two equations in Eq. (7) can be expressed as afunction of the global coordinates of point P. Reversing Eq. (1), forswitching from the local to the global coordinates, and consideringthat Ri is orthonormal – therefore R−1

i = RTi [10] – we obtain:

xi = R−1i

(X − X0i

)= RT

i

(X − X0i

). (8)

Combining Eq. (7) with Eq. (8), we obtain:

MiRTi

(X − X0i

)= 0, (9)

from which:

MiRTi X − MiR

Ti X0i

= 0. (10)

We note that the equations of this system are linear with respectto the three (unknown) coordinates of P. Eq. (10) can be expressedin compact form, as:

AiX − Bi = 0, (11)

being Ai = M iRTi and Bi = M iR

Ti X0i

.Extending Eq. (11) to the N network devices that “see” P (i.e.,

those involved in the measurement process), we define a system of2 × N equations, which can be expressed in matrix form, as:

AX − B =

⎡⎢⎢⎢⎢⎣

A1

A2

...

AN

⎤⎥⎥⎥⎥⎦ ·

⎡⎣X

Y

Z

⎤⎦−

⎡⎢⎢⎢⎢⎣

B1

B2

...

BN

⎤⎥⎥⎥⎥⎦ = 0. (12)

In practice, this system is solved to determine the unknowncoordinates of P, knowing several parameters relating to each ithnetwork device: angles �i and ϕi, subtended by P, and relevantlocation/orientation, defined by X0i

, Y0i, Z0i

, ωi, �i, and�i. Since the“true” values of the above parameters are never known exactly,


i i

from angular measurements, and X0i, Y0i

, Z0i, ωi, �i, �i, resulting

from an initial calibration process. For this reason, from this pointforward, we replace the “true” parameters with their estimates.



4 D. Maisano, L. Mastrogiacomo / Precision

Fld

tttbt

mPa

1

2

mcsPewwtuwP

ota

eerwl

awpm

ig. 3. 2D scheme displaying the different contributions to the uncertainty in theocalization of P, produced by individual network devices. For each ith networkevice, the uncertainty region is triangular.

The system in Eq. (12) can be solved when P is “seen” by at leastwo devices (2 angles × 2devices = 4 total equations). Since this sys-em is overdefined (more equations than unknown parameters),here are several possible solution approaches, ranging from thoseased on the iterative minimization of a suitable error function [8],o those based on the Least Squares method [11].

It is worth remarking that the equations of the system in Eq. (11)ay differently contribute to the uncertainty in the localization of

. Specifically, two of the main factors affecting this uncertaintyre:

. Uncertainty in angular measurements (�i and ϕi), which generallydepends on the metrological characteristics of network devices;

. Relative position between the point to be localized (P) and each ithnetwork device; in fact, assuming that the uncertainty in angularmeasurements is fixed, the uncertainty in the localization of Ptends to increase proportionally to the distance between P andnetwork devices.

The scheme in Fig. 3 makes it possible to visualize the two afore-entioned factors. For simplicity, it represents a two-dimensional

ase, in which the localization of P is performed using angular mea-urements by three network devices, with different distance from

and different angular-measurement uncertainty. Assuming thatach ith network device measures the angle (�i) subtended by Pith some uncertainty, we define a triangular uncertainty region,ith vertex in the origin (oi) of the local coordinate system, bisec-

or parallel to oiP, and vertex-angle proportional to the relevantncertainty. Obviously, the uncertainty region tends to increaseith increasing the distance between the ith network device and

.Devices that mostly contribute to uncertainty in the localization

f P – which can be roughly visualized by overlapping the uncer-ainty regions related to the three devices – are the least accuratend/or the most distant from P.

Returning to the system in Eq. (11), it can be said that the 2 × Nquations contribute to the uncertainty in the localization of P het-roskedastically with respect to the previous two factors. For thiseason, it would be appropriate to solve this system, giving greatereight to the contributions from the network devices that produce

ess uncertainty and vice versa.To this purpose, the most elegant and practical approach is prob-


bly that of the Generalized Least Squares (GLS) method [12], inhich a weight matrix, which takes into account the uncertaintyroduced by the equations of the system, is defined. One of theost practical ways to define this matrix is the application of the

PRESS Engineering xxx (2015) xxx–xxx

Multivariate Law of Propagation of Uncertainty to the system inEq. (11), referring to the parameters affected by uncertainty [13].Assuming that such parameters are the angles measured by eachith device, we collect them in a vector �:

� =[

�1, ϕ1, �2, ϕ2, . . ., �N, ϕN

]T. (13)

For simplicity, we have not taken into account the uncertaintyrelated to the estimates of the location/orientation parameters ofnetwork devices, namely the so-called external parameters, con-tained in X0i

and Ri (see Section 2), which result from calibrationprocess [5].

Propagating the uncertainty of the equations in Eq. (11), withrespect to �, we define the weight matrix as:

W =(

J · cov(

�)

· J)−1

. (14)

We describe in detail the elements in the second member of Eq.(14). J is the Jacobian (block-diagonal) matrix of the partial deriva-tives of the components of the equations in the first member of Eq.(11), with respect to the elements of �:

J =

⎡⎢⎢⎢⎢⎣

J1 0 · · · 0

0 J2 · · · 0

......

. . ....

0 0 · · · JN

⎤⎥⎥⎥⎥⎦ , (15)

where the single ith block is defined as;

Ji =[

xi · cos �i + yi · sin �i 0

zi · sin �i · cos ϕi xi cos ϕi + zi · cos �i · sin ϕi

], (16)

and 0 is a 2 × 2 matrix of zeros.We note that block Ji depends on the coordinates of P, in the local

coordinate system of the ith network device; they can be expressedas a function of the global coordinates, by applying the transforma-tion in Eq. (8). To define the elements in Ji, P has therefore to belocalized, at least roughly. One option is to use the Ordinary LeastSquares (OLS) method [14] to solve the system in Eq. (11), as:

ˆX =(

AT A)−1 · AT · B, (17)

where the “double-hat” symbol “ ” indicates that this solution isa relatively rough estimate of the final position estimate (i.e., X),which will be presented later on.

Returning to the description of Eq. (13), cov(�) is the covariancematrix of �, defined as

cov(

�)

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(2

�10

0 2ϕ1

)0 · · · 0

0

(2

�20

0 2ϕ2

)· · · 0

......

. . ....

0 0 · · ·(

2�N

0

0 2ϕN

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (18)

where 0 is a 2 × 2 matrix of zeros. The diagonal elements of cov(�)are the variances related to the angles measured by each ith device;in Section 3, we illustrate some practical ways to estimate theseparameters. The off-diagonal entries of each ith block are zeros,assuming no correlation between the �i and ϕi measurements by a


generic ith device; the off-block-diagonal entries are zeros, assum-ing that network devices work independently from each other andthere is no correlation between the �i and ϕi related to differentnetwork devices.


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ARTICLERE-6260; No. of Pages 14

D. Maisano, L. Mastrogiacomo / Pr

By applying the GLS method to the system in Eq. (11), we obtainhe final position estimate of point P as:

ˆ =(

AT · W · A)−1 · AT · W · B. (19)

Summarizing, the localization of P is based on three steps: (i)ough localization of P, using the OLS method (see Eq. (16)), (ii)onstruction of the weight matrix W (see Eq. (13)), and (iii) finalocalization of P, using the GLS method (see Eq. (18)). For furtheretails on the GLS method, see [12].

. Proposed methodology

This section is organized into three subsections: Section 3.1rovides a general description of the proposed methodology, Sec-ion 3.2 presents a practical application example, while Section 3.3resents an experimental check of the plausibility of the resultsresented in Section 3.2.

.1. General description

As mentioned in Section 1, we consider a generic distributedVM system, which includes two different types of networkevices: (i) some accurate but expensive, hereafter classified asype-A devices and (ii) other less accurate but cheaper, hereafterlassified as type-B devices.

The proposed methodology is based on three steps, which areescribed individually in Sections 3.1.1 to 3.1.3:

network layout definition;definition of the features of network devices;definition and implementation of the simulations.

Section 3.1.4 sums up the basic steps of the methodology, inhe form of pseudo-code concerning an ad hoc routine that wasmplemented in Matlab®.

.1.1. Network layout definitionThe following list contains some simplifying assumptions about

he network layout relating to the distributed LVM system ofnterest; these assumptions reflect realistic conditions in typical

orking environments, such as aerospace and rail industry work-hops.

We remark that LVM distributed systems are scalable, sinceetwork can be extended or reduced, depending on the requiredeasurement volume [5]. For this reason, the layout definition

resented in the remainder of this section is also scalable.

The network covers an appropriate measuring volume, character-ized by a surface (S) of the order of magnitude of several tens ofsquared meters and height from the floor level of about 3 m.Network devices are positioned according to a regular grid withsquare meshes (with side s), in the centre of which networkdevices are positioned (see the example in Fig. 4). The networkdensity, i.e., the number of network devices per surface unit, isdefined as:

ı = 1s2

. (20)

For example, in the network exemplified in Fig. 4,ı = 1/(2 m)2 = 0.25 devices/m2.Network devices are positioned with appropriate orientation andheight with respect to the floor level, so as to cover the largest pos-


sible measurement volume. These features may depend on thenetwork devices in use; for example, photogrammetric camerasare generally positioned at a height of approximately 4 to 5 mand oriented downwards, while optical theodolites and R-LATs


are generally positioned at a height of about 1.5 m, with verticalzi axis.

• For simplicity, we exclude the presence of obstacles betweennetwork devices and points to be localized.

• The mix between type-A and B network devices is defined usingthe following (complementary) parameters:

pA = TypeA devicesTotal network devices

,

pB = 1 − pA = TypeB devicesTotal network devices

. (21)

E.g., the network exemplified in Fig. 4 consists of 50 networkdevices, including 10 type-A devices and 40 type-B devices, there-fore pA = 20% and pB = 80%. The distribution of these devices israndom, provided that they are positioned in the predeterminedpositions, i.e., at the centre of the meshes of the grid.

• The parameters ı and pA (defined in Eqs. (20) and (21), respec-tively) are changed at multiple levels, defining a number (C) ofconfigurations. For example, having defined 4 levels for ı and 5for pA, there will be total 4 × 5 = 20 configurations.

3.1.2. Definition of the features of network devicesThe proposed procedure requires the definition of several tech-

nical/metrological features concerning network devices, as follows:

• Definition of the angular-measurement uncertainty of each ithnetwork device, in the form of standard deviations �i

and ϕi.

These parameters can be obtained in several ways: (i) from man-uals or technical documents relating to the network devices inuse, (ii) estimated through ad hoc experimental tests, or (iii)estimated using data from previous calibration processes. Weremark that the �i

and ϕivalues should reflect the network

devices’ uncertainty in realistic working conditions, e.g., in thepresence of vibrations, light/temperature variations and othertypical disturbance factors. Obviously, type-A network deviceswill be characterized by lower uncertainties, with respect to type-B ones.

• Definition of the uncertainty in the location/orientation of net-work devices, in the form of variances 2

X0i, 2

Y0i, 2

Z0i, 2

ωi, 2

�i, 2

�i.

These uncertainties may depend on several features characteriz-ing the calibration process (e.g., amount of measurements, sizeof the calibrated artefacts in use, etc.) [15].

• Definition of the range of measurement of each network device,usually expressed in terms of (i) maximum distance range(dMAX) between the network device and the point to be locatedand (ii) minimum and maximum angular range, relating tothe elevation angle (i.e., ϕMIN,ϕMAX). For example, a medium-quality photogrammetric camera has dMAX ≈ 6 to 8 m, ϕMIN ≈ 45◦

and ϕMAX ≈ 90◦; on the other hand, an R-LAT has dMAX ≈ 20 m,ϕMIN ≈ −40◦ and ϕMAX ≈ 40◦ [5].

• Definition of the purchase cost of network devices; of course, thepurchase cost of type-A cameras (i.e., CA) is likely to be higherthan that of type-B ones (i.e., CB). Combining these data with thoserelating to surface density (ı) and mix between type-A and B net-work devices (pA), we define the network’s cost per surface unit,as:

c = [pA · CA + pB · CB] · ı = [pA · CA + (1 − pA) · CB] · ı. (22)

Obviously, the total cost of a network, which covers a surfaceS, will be given by the product S × c.


3.1.3. Definition and implementation of the simulationsBased on the assumptions about network layout and the char-

acteristics of network devices (defined in Sections 3.1.1 and 3.1.2,




s = 2m

s = 2m

No. of network devices = 50 Type -A = 10 Type-B = 40 S = 200m2

s = 2m δ = 1/s2 = 0.25 d evices/m2

pA = 10/50 = 20% pB = 40/ 50 = 8 0%

Key:

twork

rwTm

rdod

liFdtd

ioneGrdserat

gt

Fig. 4. Example of ne

espectively), we now analyse the performance of different net-ork configurations, obtained by varying the parameters ı and pA.

o this purpose, we define a large number of simulated experi-ents, as follows.For each configuration, we define a number (R) of

epositionings—i.e., random assignments of (type-A and B) networkevices to the predetermined positions of the grid. Fig. 4 showsne of the possible repositionings for a configuration with ı = 0.25evices/m2 and pA = 20%.

The decision to perform multiple repositionings is aimed at ana-yzing the “average” metrological performance of not-necessarily-dentical networks, characterized by the same parameters ı and pA.or each repositioning, the “true” location/orientation of networkevices is deliberately distorted, to take account of the uncer-ainties resulting from calibration process; in formal terms, weetermine:

X0i= X0i

+ εX0ibeing εX0i

∼N(

0, 2X0i

)Y0i

= Y0i+ εY0i

being εY0i∼N(

0, 2Y0i

)Z0i

= Z0i+ εZ0i

being εZ0i∼N(

0, 2Z0i

)ωi = ωi + εωi

being εωi∼N(

0, 2ωi

)�i = �i + ε�i

being ε�i∼N(

0, 2�i

)�i = �i + ε�i

being ε�i∼N(

0, 2�i

)

, (23)

.e., the parameters related to the “true” location/orientationf each ith network device are distorted by adding zero-meanormally distributed errors, with known variances. This hypoth-sis is reasonable in the absence of systematic error causes.iven that the errors resulting from calibration are generally

elated to the technical/metrological characteristics of networkevices [15], it is reasonable to assume that devices of theame type (A or B) are characterized by the same calibrationrrors. Also, in the absence of spatial/directional effects, it iseasonable to assume that these errors are isotropic, i.e., theyre uniformly spread over the three spatial directions. In formalerms:

2X0i

= 2Y0i

= 2Z0i

={

2X0A

∀ ith device of typeA

2X0B

∀ ith device of typeB

2ωi

= 2�i

= 2�i

={

2ωA

∀ ith device of typeA

2 ∀ ith device of typeB

. (24)


ωB

Subsequently, for each of the R repositionings, we randomlyenerate a number (M) of points, distributed uniformly withinhe measurement volume. Since the true position of each point

layout (plan view).

is known a priori, it is possible to (i) identify the subset ofnetwork devices, which include the point in their range of mea-surement, and (ii) calculate the “true” angles subtended by thepoint with respect to these network devices (using the formu-lae in Eq. (3)). To take the angular-measurement uncertaintyinto account, the true angles (i.e., �i and ϕi) are deliberatelydistorted:

�i = �i + ε�ibeing ε�i

∼N(

0, 2�i

)ϕi = ϕi + εϕi

being εϕi∼N(

0, 2ϕi

) , (25)

where ε�iand εϕi

are zero-mean normally distributed randomerrors, with variances 2

�iand 2

ϕi, respectively. In the absence of

systematic error causes and spatial/directional effects, it is rea-sonable to assume that (i) for each ith network device, the twovariances are coincident, and (ii) devices of the same type are char-acterized by the same variances; in formal terms:

2�i

= 2ϕi

={

2�A

∀ ith device of type A

2�B

∀ ith device of type B. (26)

Based upon this assumption, the matrix cov(�) (in Eq. (18)) isgreatly simplified, as it includes only two parameters (i.e., 2

�Aand

2�B

), repeated depending on the number of type-A and type-B cam-eras involved in the measurement.

For each jth point related to a certain rth repositioning of net-work devices, we define N as the number of network devices thatcan “see” the point, since they include it in their range of measure-ment. If N ≥ 2, the localization can be simulated and the positionerror can be estimated as:

εPj,r=∥∥X j,r − X j,r

∥∥=√(

Xj,r − Xj,r

)2 +(

Yj,r − Yj,r

)2 +(

Zj,r − Zj,r

)2, (27)

being X j,r =[Xj,r, Yj,r, Zj,r

]Tthe true position of the point to be

located, which is known a priori, and X j,r =[Xj,r , Yj,r , Zj,r

]Tthe

localization obtained by applying the model illustrated in Section2.

If N < 2, the jth point cannot be localized. For each repositioning,we define Lr as the set including the subscripts (j) of the points thatcan be localized; the number of elements in Lr is:


mr = |Lr | ≤ M, (28)

where the symbol “|Lr|” denotes the cardinality of set Lr.Repeating the exercise for the totality of the (M) points in each

rth repositioning, we can determine the distribution of the position


ING ModelP

ecision

et

icbs

C

nw

d

a

twiidd

ε

b(a

cfic

rtsvit

c

tra


D. Maisano, L. Mastrogiacomo / Pr

rror relating to the configuration examined, as the union of allhe εPj,r

values available. The total number of realizations of εPj,r

s∑R

r=1mr ≤ R · M. We now define an indicator of the (average)overage level related to a certain network configuration, as the ratioetween the number of localized points and the overall number ofimulated points:

vg =∑R

r=1mr

R · M. (29)

Obviously, Cvg ∈ [0, 1]; this indicator will tend to decrease foretworks characterized by low density (ı) and/or network devicesith relatively small range of measurement.

Next, we calculate the following statistics, concerning the εPj,r

istribution:

�P = 1∑Rr=1mr

∑R

r=1

∑∀j∈ Lr

εPj,r

P =

√√√√√∑R

r=1

∑∀j∈ Lr

(εPj,r

− �P

)2(∑Rr=1mr

)− 1

, (30)

Due to the definition of εPj,r, the two indicators �P and P will

lways be larger than or equal to 0.Data resulting from simulations can be also used for estimating

he uncertainty relating to distance measurements. To this purpose,e consider all the possible pairs of points in each reposition-

ng; since the number of localized points in a certain repositionings mr, the total number of possible pairs of points (and relevantistances) will be Cmr

2 = (mr!) / (2! (mr − 2)!). We can define theistance error as:

djk,r= djk,r − djk,r =

∥∥X j,r − Xk,r

∥∥−∥∥X j,r − Xk,r

∥∥ , (31)

eing X j,r and Xk,r the estimated positions of the jth and kth pointwhere j and k ∈Lr); Xj,r and Xk,r the relevant true positions (known

priori).Considering the totality of pairs for all the R repositionings, we

an determine the distribution of the distance error, for the con-guration of interest, as the union of all the εdjk,r

values. Next, wealculate the following statistics:

�d = 1R · Cmr

2

∑R

r=1

∑∀j∈ Lr

∑∀k∈Lr : k>j

εdjk,r

d =

√√√√√∑R

r=1

∑∀j∈ Lr

∑∀k∈Lr : k>j

(εdjk,r

− �d

)2(∑Rr=1Cmr

2

)− 1

,

(32)

Due to the definition of εdjk,r, the indicator �d is expected to be

oughly 0, while d ≥ 0. Between �d and d, d is the most represen-ative statistic for distance-measurement uncertainty. Also, usingd values is more practical than using �P or P values; the rea-on is that typical industrial applications concern the dimensionalerification of distances between pairs of points on the object ofnterest (e.g., point clouds depicting the object’s surface) and nothe absolute position of these points.

Finally, the cost per surface unit (c) relating to each configurationan be estimated using Eq. (22).


The so-far-described procedure should be extended to theotality of the network configurations. The large amount of dataesulting from this analysis can be organized into appropriate tablesnd graphs, which are illustrated in Section 3.2.


3.1.4. Ad hoc routineThe methodology illustrated in the previous three sub-sections

has been automated by an ad hoc routine developed in Matlab®;the following pseudo-code summarizes the basic steps of thisroutine:

Network Layout definition1. Define the features of the measuring volume (e.g., surface,

maximum height from the floor level, etc.).2. Define the levels of ı and pA , which characterize the (C)

configurations.Definition of the features of the network devices3. For each of the two types (A and B) of network devices:4. Set the uncertainty in angular measurements.5. Define their orientation/height with respect to the floor level.6. Set the uncertainty in location/orientation, resulting from

calibration.7. Set the typical range of measurement.8. Set the unitary cost.9. End For.Definition and implementation of the simulations10. For each of the C configurations:11. Determine the cost per surface unit (c), using Eq. (22).12. Define the number of repositionings (R) of the network devices.13. For each rth of the R repositionings:14. Randomly assign type-A and B network devices to the

predetermined positions (on the grid).15. Distort the true location/orientation of each network device, in

order to take account of the uncertainties resulting fromcalibration (Eq. (23)).

16. Define M points, randomly distributed over the measuringvolume.

17. For each of the M points:18. Determine the portion of N devices, which can “see” the

point.19. If N ≥ 2:20. For each of these N devices:21. Calculate the true �i and ϕi angles, using Eq. (3).22. Distort the true angles, in order to take account of the

angular-measurement uncertainty (Eq. (25)).23. End For.24. Localize the point, applying the mathematical model in

Eq. (19).25. Calculate the position error (Eq. (27)).26. End if.27. End For.28. For each of the possible (Cmr

2 ) pairs of points:29. Determine the “true” and estimated distance, then the

corresponding distance error (Eq. (32)).31. End For.32. End For.33. Determine the indicator Cvg, depicting the coverage level (Eq.

(29)).34. Perform the union of the (available) position errors and

determine the statistics in Eq. (30).35. Perform the union of the (available) distance-measurement

errors and determine the statistics in Eq. (32).36. End For.37. Generate appropriate tables and charts, representing the results

obtained.38. End.

3.2. Application example

This section presents a practical application of the methodol-ogy described in Section 3.1, to design a multi-sensor networkfor a low-end distributed LVM system, based on infrared (IR)photogrammetry. Before going into the description of the appli-cation, Section 3.2.1 makes a brief digression to recall somebasic concepts about photogrammetric systems in general. Sec-


tion 3.2.2 focuses on the definition of the network layout, thefeatures of network devices, and the simulated experiments.Section 3.2.3 presents and comments the results of the simula-tions.



8 D. Maisano, L. Mastrogiacomo / Precision

Fig. 5. For a generic ith network camera, representation of the local coordinatesp

3

awtat

gafm

amPxtF

�

2<

− �

2�

2

wtott

3

i

1

2

network cameras are deliberately distorted, as described in Eq. (23).

ystem, with origin (oi) in the projection centre (or focus), and the image plane uivi ,arallel to the plane xiyi , at a distance fi (i.e., the focal length).

.2.1. Basic concepts on photogrammetric systemsFor most of the existing systems, network devices are IR cameras

ssociated with IR illuminators, while targets are reflective spheres,hose centres should be localized. Reflective spheres are passive

argets illuminated by the illuminators. Alternatively, one can usective spherical targets that emit IR light, not making it necessaryo use illuminators.

The measurement uncertainty of photogrammetric systemsenerally depends on the cameras in use: high pixel resolutionnd good quality lens help in reducing measurement uncertaintyor static measurements, while high frame rate helps in reducing

easurement uncertainty for dynamic measurements.We now focus the attention on each ith network camera. The

ngles �i and ϕi are not measured directly: each ith network cameraeasures the coordinates of P′′ (ui, vi), i.e., the projection of target

on the camera’s image plane uivi, which is parallel to the planeiyi of the local coordinate system. Knowing these coordinates andhe camera focal length (fi), it is possible to estimate �i and ϕi (seeig. 5):

�i = tan−1vi − v0i

ui − u0i

⎧⎪⎨⎪⎩

if ui − u0i> 0 then

if ui − u0i≤ 0 then

ϕi = tan−1 fi√(ui − u0i

)2 +(vi − v0i

)2{

−�

2≤ ϕi ≤

here ui and vi are the coordinates of the projection (P′′) of P onhe image plane; u0i

and v0iare the coordinates of the projection of

i on the image plane; fi is the distance between the plane uivi andhe camera projection centre (or focus), which is coincident withhe origin oi of the local coordinate system oixiyizi.

.2.2. Definition of the simulationsWe assume that the photogrammetric system of interest

ncludes two types of network cameras:

. Type-A: Hitachi, Gigabit Ethernet Infrared cameras, with


1360 × 1024 pixel resolution and 30 fps;. Type-B: PixArt/WiiMote Infrared cameras, with 126 × 96 pixel

resolution and 100 fps.


�i <3�

2

≤ �i ≤ �

2 , (33)

It is worth noting that, while type-A cameras are professionalproducts of good quality, type-B cameras can be classified as toycameras.

The list below reports some assumptions about the layout ofnetwork cameras and the measurement volume.

• Network cameras are positioned on the ceiling, at 4.5 m in heightfrom the floor, facing down and positioned at the centre of thesquare meshes (with side s) of the grid;

• The parameter ı (or s) is changed on 14 levels (i.e., s = 0.75, 1, 1.25,1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 3.75 and 4 m), while param-eter pA on 10 levels (i.e., pA = 0%, 2.5%, 5%, 7.5%, 10%, 12.5%, 15%,17.5%, 20%, 22.5%), for total C = 14 × 10 = 140 configurations. Foreach configuration, we define R = 200 random repositionings ofnetwork devices. For each rth repositioning, we generate M = 100points (to be localized), distributed uniformly in the measure-ment volume. Among these M points, those that can be localized– since they are within the range of measurement of at least twonetwork devices –are mr ≤ M. The Cmr

2 possible pairs of pointsdetermine as many corresponding distances. In the case of per-fect coverage, i.e., when all the M points in each repositioningcan be localized, the total number of points and distances avail-able are, respectively, R × M = 200 × 100 = 20,000 and R × C100

2= 200 × 4950 = 990,000 for a generic configuration. The measure-ment volume (which includes the randomly generated points)is a parallelepiped region with surface S = 15 × 15 = 225 m2 andheight from the floor level of 3 m. This way, we exclude “low cov-erage” regions, i.e., regions at relatively large height, which arepoorly covered by the range of measurement of network cam-eras; details on the range of measurement of network deviceswill be given in the next paragraph.

Table 1 reports the technical/metrological features of the net-work devices in use. Data concerning the angular-measurementuncertainty are obtained through preliminary experimental tests.Uncertainties relating to the location/orientation of networkdevices can be obtained from previous calibration processes; thefact that the uncertainties related to type-A cameras are generallylower than those related to type-B cameras is a consequence of thelower angular-measurement uncertainty [16].

The range of measurement of each ith device is a cone with axisparallel to zi axis (of the local coordinate system), vertex at the ori-gin oi, vertex-angle 2 × (90◦ − ϕMIN), surmounting a spherical capcentred in oi and with radius dMAX (see Fig. 6). Knowing the rangeof measurement of network devices, we can determine, for each ofthe simulated points, the set of (N) cameras involved in the mea-surement and then determine whether the point can be localized(i.e., N ≥ 2).

For each of the R repositionings, the true position/orientation of


For each of the M points, the true angles related to each device aredistorted too, as described in Eq. (25). Next, when N ≥ 2, localizationis simulated and position error is calculated using Eq. (27). For each



D. Maisano, L. Mastrogiacomo / Precision

Table 1Technical/metrological features of the (type-A and B) network cameras..

Type-A camera Type-B camera

Model Hitachi, GigabitEthernet

PixArt/WiiMote

Pixel resolution 1360 × 1024 126 × 96Frame rate ≈30 fps ≈100 fpsLight type Infrared InfraredUncertainty in angular

measurement(

�i

) ≈0.02◦ ≈0.3◦

Location uncertainty(X0i

), from

calibration

≈0.08 mm ≈1.1 mm

Orientationuncertainty

(ωi

),

from calibration

≈0.02◦ ≈0.1◦

Range of measurementϕMIN ≈40◦ ≈45◦

ϕMAX ≈90◦ ≈90◦

dMAX ≈8 m ≈6 mPurchase cost ≈350 D ≈40 D

cot

trot

pr

3

tItontcm

thnt

calibrated artefacts, within the measurement volume [16].Next, we perform some dimensional measurements using three

calibrated bars of different lengths: about 1 m, 1.5 m and 2 m (see

Fig. 6. Range of measurement of a generic (ith) photogrammetric camera.

onfiguration, we therefore have several thousands of realizationsf the position error, forming a distribution for which we determinehe statistics in Eq. (30).

Considering the totality of the pairs of points in each configura-ion, it is possible to estimate the relevant distance errors, using theelationship in Eq. (31). We therefore have hundreds of thousandsf realizations of the distance-measurement error, forming a dis-ribution for which we determine the statistics in Eq. (32).

Finally, for each configuration, the Cvg indicator and the coster surface unit are determined by applying Eqs. (29) and (22),espectively.

.2.3. ResultsTable A.1 (in Appendix A) summarizes the results of the simula-

ions. Each row is related to one of the 140 configurations examined.t can be noticed that the statistics relating to measurement uncer-ainty (e.g., �P, P, �d, and d) are relatively high. i.e., in the orderf magnitude of a few mm. This is due to the fact that the modelledetwork includes two types of cameras with a very clear gap inerms of metrological performance; given that the majority of theameras (i.e., those of type-B) are not very accurate, the resultingeasurement uncertainty will be inevitably large.Not surprisingly, the higher the parameters ı and pA, the lower


he measurement uncertainty (both in terms of P and d) and theigher c. In addition, the network configurations with relatively lowetwork density are the ones with the poorest coverage; reversinghe perspective, it can be noticed that the first 40 configurations


in Table A.1 (in Appendix A) have Cvg ≥ 95%, due to their relativelyhigh ı values.

Fig. 7 displays the results in a more intuitive way, through two3D surfaces (and relevant contour lines) representing, respectively,d and c as functions of both ı and pA. Also, in the graphs containingthe contour lines, we identify three regions with different coveragelevels: respectively Cvg < 95%, 95% ≤ Cvg < 99%, and Cvg ≥ 99%.

These graphs show that a good strategy to achieve a certaind (and Cvg), without making c increase excessively, is selectinga sufficiently high pA and reducing ı to the necessary extent, con-sistently with the requested d. For example, assuming that therequired d ≤ 5 mm and Cvg ≥ 95%, the network configuration thatminimizes c is the one with ı = 0.44 devices/m2 and pA = 17.5% (seethe configuration no. 33 in Table A.1).

The same result can be obtained looking through the maps inFig. 8, obtained by superimposing the iso-d contour lines (i.e.,d = constant) with iso-c (i.e., c = constant). This graph also showsthat, as expected, low uncertainties are often incompatible withlow costs.

In the previous graphs, d (not P) is used as indicator of mea-surement uncertainty. The fact remains that d and P are stronglycorrelated by an approximately linear relationship (e.g., R2 ≈ 95%considering the data in Table A.1).

3.3. Preliminary experiments

To check the plausibility of the results presented in Section3.2, we perform some experiments on a low-end IR photogram-metric system. This system has been developed at the industrialand quality engineering laboratory of DIGEP—Politecnico di Torinoand includes the same two types of network cameras described inTable 1.

It is assumed that (i) floor area of the measurement volumeis S ≈ 6 × 5 = 30 m2, (ii) required uncertainty on distance measure-ment is d ≈ 2.5 mm and (iii) total budget is 3000D . The maximumcost per surface unit is therefore c = 3000D /30 m2 ≈ 100D /m2.

Table A.1 shows that the cheapest configuration, compatiblewith the required d is that with ı = 1 device/m2 (or s = 1 m) andpA = 15% (see configuration no. 14 in Table A.1); for this config-uration c = 86.5D /m2 and Cvg ≈ 100%. Interestingly, the budget isalso compatible with a configuration with a d even lower thanthe required one: in fact, configuration no. 13 in Table A.1 – withı = 1 device/m2 (or s = 1 m) and pA = 17.5% – is associated withd ≈ 2.21 mm and c ≈ 94.3D /m2.

Adopting the latter configuration, we set a network withS × ı = 30 total cameras, among which pA × S × ı ≈ [5.25] = 5 of type-A and pB × S × ı ≈ [24.75] = 25 of type-B. We remark that, afterrounding the number of type-A and B cameras to the nearest inte-ger, there is a slight reduction in c, from 94.3D /m2 to 91.7D /m2

(calculated using Eq. (22)). The resulting total cost of the networkis therefore S × ı ≈ = 2750D . Also, there is a slight reduction in pA,from 17.5% to 5/30 ≈ 16.7%.

Cameras are arranged at the centre of square meshes with sides ≈ 1 m of a regular grid, at a 4.5 m height from the floor (in line withthe assumption of the model used in Section 3.2). Type-A and B cam-eras are distributed relatively uniformly over the predeterminedpositions3; Fig. 9 contains a scheme of the experimental networklayout. The location/orientation of network devices are determinedthrough a calibration process, based on multiple measurements of


3 In general, a uniform distribution of type-A cameras helps in limiting areas withrelatively high uncertainty, since they are exclusively covered by type-B cameras.




F (in Apv 5%, (ii

FCt2u

temcn

Fl

ig. 7. 3D surfaces and corresponding contour lines related to the data in Table A.1alues. Regarding contour lines, the pA-ı plan is divided in three regions: (i) Cvg < 9

ig. 10). The true values of these lengths were measured with aoordinate Measuring Machine (CMM) with measurement uncer-ainty of the order of magnitude of a few tenths of �m, i.e., at least–3 orders of magnitude lower than the (purported) measurementncertainty of the photogrammetric system in use [17].

For each of the three calibrated bars, we place two targets athe two ends of the bar and perform 50 measurements, in differ-


nt positions within the measurement volume. Specifically, in eacheasurement, targets are localized and the length of the bar is cal-

ulated as Euclidean distance between the two targets. The totalumber of distance measurements is therefore 150.

ig. 8. Map representing the iso-d (i.e., d = constant) and iso-c contour lines (i.e., c = coines, while figures in the top side refer to iso-d contour lines.

pendix A). The figures on surfaces and contour lines, refer to the relevant d and c) 95% ≤ Cvg < 99%, and (iii) Cvg ≤ 99%.

The distance error is defined as:

εdjk= djk − djk, (34)

being djk is the distance calculated using the relationship∥∥X j − Xk

∥∥, where X j and Xk are the estimated positions of the twotargets; djk is the “true” value of djk (determined using the CMM).


Aggregating the 150 distance-measurement errors, we con-struct the experimental distribution of the εdjk

values anddetermine the statistics reported in Eq. (32). These statistics arecompared with those obtained from simulations, for a similar

nstant) relating to Fig. 7. The figures in the bottom/right side refer to iso-c contour



D. Maisano, L. Mastrogiacomo / Precision Engineering xxx (2015) xxx–xxx 11

No. of network de vices = 30 Type -A = 5 Type -B = 25 S = 6·5 = 30m2

s = 1m δ = 1/s2 = 1 device /m2

pA = 5/30 = 16.7% pB = 25/30 = 83.3% CA = 350€ CB = 40€ c = [pA·CA + pB·CB]·δ = 91.7€ Total cost = S·c = 2,75 0€

s = 1m

s = 1m

Key:

6m

5m

Fig. 9. Scheme of the experimental network layout (plan view).

Table 2Comparison between the results obtained from simulated and actual experiments, for a certain network configuration.

ı [m−2] s [m] pA [%] c [D /m2] Cvg [%] � [mm] [mm]

Simulations 1 1 17.5

Experiments 1 1 16.7

ci

cpeups

4

nBtc

rta

simulations, with no more need to estimate them empirically.

Fig. 10. Photo of one of the calibrated bars used for the experiments.

onfiguration (i.e., the statistics concerning configuration no. 13,n Table A.1) (Table 2).

The agreement between these data is comforting, since it indi-ates that the proposed methodology is feasible and provideslausible results. In particular, it can be noticed that the dstimated experimentally is very close to that resulting from sim-lations. This result is also remarkable, in light of the fact that theA used in the experiments is slightly lower than that used in theimulations.

. Concluding remarks

This paper proposed a methodology to support the design ofetworks for distributed LVM systems based on triangulation.ased on a large number of simulations, this methodology allowso identify the most appropriate network configuration(s), which isompatible with the required measurement uncertainty and cost.


The proposed methodology was automated through an ad hocoutine, developed in Matlab®, which makes it possible to simulatehe performance of a number of possible network configurationsnd construct tables and charts, which help in identifying the most

d d

94.3 100.0 0.03 2.2191.7 100.0 0.07 2.15

convenient one(s). Furthermore, simulations allow to estimate thecoverage level of a network and, consequently, to avoid adoptingconfigurations with poor coverage.

The mathematical model for localization is based on the GLSmethod and is well suited to multi-sensor networks, i.e., networksin which devices of different nature can coexist.

Through some experiments carried out at Politecnico diTorino—DIGEP, we checked the plausibility of the results of themethodology, for a specific distributed LVM system with two typesof photogrammetric cameras. These experiments confirmed thefeasibility and practicality of the proposed methodology.

The methodology has some limitations, summarized as follows:

• It requires several characteristic data relating to network devices(e.g., uncertainty in angular measurements, range of measure-ment, location/orientation uncertainty, etc.), which – if notavailable – should be determined experimentally by ad hoc tests.

• The methodology cannot be applied to multi-sensor networkswith more than two types of devices and/or devices able to per-form distance measurements.

• The mathematical model used for localization weights the uncer-tainty contributions from the different network devices, takinginto account (i) their uncertainty in angular measurements and(ii) the relative position/orientation of these devices with respectto the target; however, the model does not take into account theuncertainty in their location/orientation.

Future research is aimed at generalizing the proposed method-ology, in order to extend it to LVM distributed systems withnetwork devices able to perform both angular and distance mea-surements. Also, we plan to improve the methodology, in order tosimulate the network calibration and determine the uncertaintiesin the location/orientation of network devices through appropriate


Appendix A.

See Table A.1.


Please cite this article in press as: Maisano D, Mastrogiacomo L. A new methodology to design multi-sensor networks for distributedlarge-volume metrology systems based on triangulation. Precis Eng (2015), http://dx.doi.org/10.1016/j.precisioneng.2015.07.001



Table A.1Results of the simulations relating to the application example. Configurations are arranged in (descending) lexicographic order with respect to ı and pA , respectively. Theconfigurations highlighted in grey are characterized by a relatively low coverage (i.e., Cvg < 95%).

Config. no. ı [m−2] s [m] p [%] c [D /m2] Cvg [%] �d [mm] d [mm] �P [mm] P [mm]

1 1.78 0.75 22.5 195.1 100.0 0.00 1.18 1.15 0.922 1.78 0.75 20.0 181.3 100.0 0.00 1.26 1.22 0.923 1.78 0.75 17.5 167.6 100.0 −0.01 1.36 1.32 0.994 1.78 0.75 15.0 153.8 100.0 −0.02 1.45 1.43 1.105 1.78 0.75 12.5 140.0 100.0 −0.02 1.75 1.59 1.336 1.78 0.75 10.0 126.2 100.0 0.00 1.85 1.78 1.477 1.78 0.75 7.5 112.4 100.0 0.03 2.22 2.08 1.708 1.78 0.75 5.0 98.7 100.0 0.04 2.55 2.51 1.999 1.78 0.75 2.5 84.9 100.0 0.04 3.25 3.22 2.3610 1.78 0.75 0.0 71.1 100.0 0.14 4.64 4.76 3.1011 1.00 1.00 22.5 109.8 100.0 0.01 1.89 1.67 1.5912 1.00 1.00 20.0 102.0 100.0 0.01 1.98 1.76 1.6213 1.00 1.00 17.5 94.3 100.0 0.03 2.21 1.93 1.8114 1.00 1.00 15.0 86.5 100.0 0.01 2.46 2.14 2.0415 1.00 1.00 12.5 78.8 100.0 0.03 2.64 2.41 2.2316 1.00 1.00 10.0 71.0 100.0 0.02 2.94 2.67 2.4017 1.00 1.00 7.5 63.3 100.0 0.00 3.32 3.14 2.7318 1.00 1.00 5.0 55.5 100.0 0.03 3.98 3.78 3.1019 1.00 1.00 2.5 47.8 100.0 0.09 4.76 4.79 3.5320 1.00 1.00 0.0 40.0 100.0 0.14 6.07 6.40 4.0821 0.64 1.25 22.5 70.2 99.5 0.05 3.13 2.39 2.8822 0.64 1.25 20.0 65.3 99.6 0.01 3.05 2.63 2.9923 0.64 1.25 17.5 60.3 99.6 0.01 3.55 2.79 3.1624 0.64 1.25 15.0 55.4 99.6 0.00 3.75 3.15 3.5025 0.64 1.25 12.5 50.4 99.6 0.02 4.03 3.46 3.4726 0.64 1.25 10.0 45.4 99.7 0.06 4.64 3.92 3.9827 0.64 1.25 7.5 40.5 99.6 0.07 5.19 4.50 4.2028 0.64 1.25 5.0 35.5 99.7 0.08 5.76 5.39 4.6529 0.64 1.25 2.5 30.6 99.8 0.11 6.64 6.58 4.8530 0.64 1.25 0.0 25.6 99.7 0.13 7.82 8.13 5.2031 0.44 1.50 22.5 48.8 97.1 0.04 4.32 3.25 3.8732 0.44 1.50 20.0 45.3 97.2 0.04 4.56 3.51 4.0333 0.44 1.50 17.5 41.9 97.4 0.02 4.80 3.81 4.3034 0.44 1.50 15.0 38.4 97.6 0.04 5.22 4.30 4.7635 0.44 1.50 12.5 35.0 97.6 0.00 5.28 4.80 4.9936 0.44 1.50 10.0 31.6 97.5 0.06 5.93 5.32 5.1237 0.44 1.50 7.5 28.1 97.7 0.02 6.60 6.09 5.4538 0.44 1.50 5.0 24.7 97.9 0.09 7.30 7.14 5.8039 0.44 1.50 2.5 21.2 97.8 0.09 8.54 8.42 6.1840 0.44 1.50 0.0 17.8 98.2 0.21 9.47 9.85 6.2441 0.33 1.75 22.5 35.8 92.7 0.05 5.30 4.32 4.9842 0.33 1.75 20.0 33.3 92.6 0.07 5.88 4.71 5.5143 0.33 1.75 17.5 30.8 93.2 0.05 6.22 5.05 5.5544 0.33 1.75 15.0 28.2 92.8 0.04 6.35 5.52 5.6445 0.33 1.75 12.5 25.7 93.1 0.11 6.77 6.18 6.1046 0.33 1.75 10.0 23.2 93.6 0.13 7.69 6.82 6.2847 0.33 1.75 7.5 20.7 93.6 0.05 8.32 7.78 6.7248 0.33 1.75 5.0 18.1 94.1 0.09 9.08 8.77 6.9949 0.33 1.75 2.5 15.6 94.0 0.22 10.08 10.09 7.1350 0.33 1.75 0.0 13.1 93.9 0.29 11.12 11.63 7.2051 0.25 2.00 22.5 27.4 86.7 0.06 6.16 5.03 5.9152 0.25 2.00 20.0 25.5 86.4 0.00 6.83 5.57 6.4053 0.25 2.00 17.5 23.6 87.1 −0.01 7.08 5.88 6.3154 0.25 2.00 15.0 21.6 87.5 −0.01 7.67 6.70 6.7755 0.25 2.00 12.5 19.7 88.0 0.14 8.07 7.28 7.0656 0.25 2.00 10.0 17.8 87.7 0.12 9.18 8.24 7.4657 0.25 2.00 7.5 15.8 88.2 0.13 9.35 8.89 7.4658 0.25 2.00 5.0 13.9 88.3 0.14 10.60 10.35 8.0859 0.25 2.00 2.5 11.9 88.7 0.26 11.32 11.32 8.0660 0.25 2.00 0.0 10.0 89.0 0.23 12.46 13.25 8.0661 0.20 2.25 22.5 21.7 80.9 0.14 7.59 6.19 7.0862 0.20 2.25 20.0 20.1 81.0 0.03 8.12 6.73 7.3363 0.20 2.25 17.5 18.6 81.0 0.08 8.74 7.41 7.8064 0.20 2.25 15.0 17.1 81.7 0.13 9.27 8.03 7.9065 0.20 2.25 12.5 15.6 81.9 0.16 9.93 8.84 8.2266 0.20 2.25 10.0 14.0 81.7 0.01 10.54 9.85 8.5667 0.20 2.25 7.5 12.5 82.3 0.19 11.43 10.92 8.9168 0.20 2.25 5.0 11.0 82.9 0.26 12.18 12.09 9.0469 0.20 2.25 2.5 9.4 82.9 0.25 13.19 13.36 9.1170 0.20 2.25 0.0 7.9 83.0 0.26 14.24 14.97 9.0271 0.16 2.50 22.5 17.6 74.4 0.10 9.25 7.56 8.3072 0.16 2.50 20.0 16.3 75.0 0.09 9.31 7.93 8.4773 0.16 2.50 17.5 15.1 75.4 0.16 9.95 8.84 8.8274 0.16 2.50 15.0 13.8 76.0 0.17 10.29 9.24 8.7775 0.16 2.50 12.5 12.6 76.3 0.19 11.30 10.37 9.56



D. Maisano, L. Mastrogiacomo / Precision Engineering xxx (2015) xxx–xxx 13

Table A.1 (Continued)

76 0.16 2.50 10.0 11.4 76.4 0.09 12.55 11.82 9.7577 0.16 2.50 7.5 10.1 76.9 0.35 13.31 12.58 10.2178 0.16 2.50 5.0 8.9 76.9 0.26 13.81 13.96 9.9179 0.16 2.50 2.5 7.6 78.0 0.37 14.52 14.85 10.0380 0.16 2.50 0.0 6.4 78.0 0.39 16.00 16.84 9.9781 0.13 2.75 22.5 14.5 68.6 0.22 10.71 9.12 9.7582 0.13 2.75 20.0 13.5 68.5 0.16 11.34 9.79 10.0483 0.13 2.75 17.5 12.5 68.6 0.11 11.73 10.36 10.2484 0.13 2.75 15.0 11.4 69.5 0.13 12.72 11.59 10.6685 0.13 2.75 12.5 10.4 69.6 0.11 13.66 12.52 10.9986 0.13 2.75 10.0 9.4 70.4 0.28 14.28 13.47 11.2387 0.13 2.75 7.5 8.4 69.9 0.20 14.64 14.21 11.2188 0.13 2.75 5.0 7.3 71.0 0.42 16.05 16.43 11.5189 0.13 2.75 2.5 6.3 70.7 0.47 17.22 17.59 11.5190 0.13 2.75 0.0 5.3 71.0 0.32 17.95 18.96 11.3691 0.11 3.00 22.5 12.2 63.4 0.09 11.52 9.83 10.3392 0.11 3.00 20.0 11.3 63.0 0.15 11.72 10.39 10.6493 0.11 3.00 17.5 10.5 63.7 0.25 12.34 11.11 10.7194 0.11 3.00 15.0 9.6 64.3 0.18 13.79 12.61 11.2695 0.11 3.00 12.5 8.7 64.8 0.35 14.49 13.67 11.6296 0.11 3.00 10.0 7.9 65.1 0.46 14.92 14.47 11.6797 0.11 3.00 7.5 7.0 65.5 0.32 15.48 15.39 11.8898 0.11 3.00 5.0 6.2 66.7 0.46 17.40 17.81 11.9499 0.11 3.00 2.5 5.3 66.5 0.32 18.65 19.06 12.05100 0.11 3.00 0.0 4.4 66.3 0.60 19.10 20.37 12.05101 0.09 3.25 22.5 10.4 56.4 0.29 14.04 12.12 12.32102 0.09 3.25 20.0 9.7 56.7 0.28 14.58 13.02 12.30103 0.09 3.25 17.5 8.9 56.4 0.20 15.32 13.85 12.66104 0.09 3.25 15.0 8.2 57.7 0.52 16.12 15.24 13.16105 0.09 3.25 12.5 7.5 57.7 0.34 16.11 15.11 12.99106 0.09 3.25 10.0 6.7 58.9 0.21 17.00 16.53 13.32107 0.09 3.25 7.5 6.0 58.3 0.33 18.24 17.89 13.49108 0.09 3.25 5.0 5.3 58.7 0.60 19.26 19.43 13.59109 0.09 3.25 2.5 4.5 59.3 0.64 20.17 20.94 13.44110 0.09 3.25 0.0 3.8 60.1 0.61 21.41 22.71 13.58111 0.08 3.50 22.5 9.0 50.8 0.53 14.45 12.64 12.49112 0.08 3.50 20.0 8.3 51.3 0.39 15.35 13.83 12.85113 0.08 3.50 17.5 7.7 52.3 0.46 15.90 14.81 13.01114 0.08 3.50 15.0 7.1 52.5 0.54 17.17 16.18 13.34115 0.08 3.50 12.5 6.4 53.5 0.33 16.84 15.90 13.24116 0.08 3.50 10.0 5.8 54.1 0.36 18.00 17.42 13.73117 0.08 3.50 7.5 5.2 54.1 0.47 18.49 18.80 13.81118 0.08 3.50 5.0 4.5 54.9 0.58 20.25 20.37 13.77119 0.08 3.50 2.5 3.9 55.2 0.36 21.21 22.02 13.77120 0.08 3.50 0.0 3.3 55.7 0.65 22.47 23.82 13.77121 0.07 3.75 22.5 7.8 45.5 0.38 15.49 13.85 13.35122 0.07 3.75 20.0 7.3 45.9 0.49 16.24 14.77 13.46123 0.07 3.75 17.5 6.7 46.7 0.53 17.70 16.31 14.20124 0.07 3.75 15.0 6.2 47.2 0.53 18.59 17.81 14.54125 0.07 3.75 12.5 5.6 47.9 0.48 18.34 17.48 14.35126 0.07 3.75 10.0 5.0 48.4 0.37 19.43 18.69 14.62127 0.07 3.75 7.5 4.5 48.4 0.63 20.38 20.37 14.75128 0.07 3.75 5.0 3.9 49.0 0.89 21.31 21.87 14.77129 0.07 3.75 2.5 3.4 49.6 0.78 22.80 23.90 14.96130 0.07 3.75 0.0 2.8 50.0 0.78 23.64 25.51 14.60131 0.06 4.00 22.5 6.9 38.3 0.36 17.64 15.71 15.02132 0.06 4.00 20.0 6.4 39.3 0.68 18.33 17.04 15.24133 0.06 4.00 17.5 5.9 39.2 0.67 19.76 18.98 15.54134 0.06 4.00 15.0 5.4 39.8 0.70 19.62 18.85 15.58135 0.06 4.00 12.5 4.9 40.9 0.77 21.30 20.73 15.77136 0.06 4.00 10.0 4.4 41.2 0.47 20.97 20.87 15.82137 0.06 4.00 7.5 4.0 41.3 0.91 22.30 22.75 15.97138 0.06 4.00 5.0 3.5 41.7 0.86 24.27 25.16 15.99

R

139 0.06 4.00 2.5 3.0

140 0.06 4.00 0.0 2.5

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